## Alignment- and reference-free phylogenomics

## with colored de-Bruijn graphs

### Roland Wittler

Genome Informatics, Faculty of Technology, Bielefeld University, Germany Center for Biotechnology, Bielefeld University, Germany

roland.wittler@uni-bielefeld.de May 15, 2019

Abstract

We present a new whole-genome based approach to infer large-scale phylogenies that is alignment- and reference-free. In contrast to other methods, it does not rely on pairwise comparisons to determine distances to infer edges in a tree. Instead, a colored de-Bruijn graph is constructed, and information on common subsequences is extracted to infer phylogenetic splits. Application to different datasets confirms robustness of the approach. A comparison to other state-of-the-art whole-genome based methods indicates comparable or higher accuracy and efficiency.

### 1

### Introduction

A common task in comparative genomics is the reconstruction of the evolutionary

re-lationships of species or other taxonomic entities, their phylogeny. Today’s wealth of

available genome data enables large-scale comparative studies, where phylogenetics is faced with the following problems: First, the sequencing procedure itself is becoming cheaper and faster, but finishing a genome sequence remains a laborious step. Thus, more and more genomes are published in an unfinished state, i.e., only assemblies (com-posed of contigs), or raw sequencing data (com(com-posed of read sequences) are available. Hence, traditional approaches for phylogenetic inference can often not be applied, be-cause they are based on the identification and comparison of marker sequences, which relies on computing multiple alignments, an NP-hard task. Second, the low sequencing cost allow new large-scale studies of certain niches and/or aloof from model organisms, where reference sequences would be too distant or not available at all.

Whole-genome approaches solve these problems as they are usually alignment- and reference-free, see e.g. [4, 6, 11, 16, 17, 20]. However, the sheer number of genomes to be analysed is still posing limits in large-scale scenarios as almost all whole-genome approaches are based on a pairwise comparison of some characteristics of the genomes

(e.g. occurrences or frequencies of k-mers or other patterns) to define distances which

are then used to reconstruct a tree (e.g. by using neighbor joining [13]). This means, for

ngenomes,O(n2_{) comparisons are performed in order to infer}_{O}_{(}_{n}_{) edges. To the best}

of our knowledge, only MultiSpaM [3] follows a different approach by sampling small,

gap-free alignments involvingfour genomes each, which are used to infer a super tree

on quartets. According to our experiments, this method is not suitable for large-scale settings (see Results), though.

Apart from computational issues, the actual objective of phylogenetic inference in terms of how to represent a phylogeny is not obvious in the first place. Taking only

intra-genomic mutations into account, i.e., assuming a genome mutating independently of others, genomes would have unique lines of ancestors and their phylogeny would thus be a tree. Several reasons however conflict this simple tree model. Inter-genomic exchange of genomic segments such as crossover in diploid or polyploid organisms, lateral gene transfer in bacteria, or introgression in insects contradict the assumption of unique

ancestry. Furthermore, incomplete, ambiguous, or even misleading information can

hamper resolving a reliable phylogenetic tree.

Here, we propose a new methodology that is whole-genome based, alignment- and reference-free, and does not rely on a pairwise comparison of the genomes or their char-acteristics. An implementation called SANS (”Symmetric Alignment-free phylogeNomic

Splits“) is available athttps://gitlab.ub.uni-bielefeld.de/roland.wittler/sans.

The k-mers of all genomic sequences (assemblies or reads) are stored in a colored

de-Bruijn graph, which is then traversed to extract phylogenetic signals. The reconstructed

phylogenies are not restricted to trees. Instead, the generalized model of phylogenetic

splits [2] is used to infer phylogenetic networks that can indicate a tree structure and

also point to ambiguity in the reconstruction.

In the following Section 2, we will first introduce two building blocks of our approach,

splits andcolored de-Bruijn graphs. Then, we will describe and motivate our method in

Section 3. After an evaluation on several real data sets in Section 4, we will give a brief summary and an outlook in Section 5.

### 2

### Background

Before presenting our method in Section 3, we will introduce two basic concepts it builds

upon. Firstly, as motivated above, our phylogenies will be represented by sets ofsplits, a

generalization of trees. Secondly, to extract phylogenetic signals from the given genomes

at the first place, they are stored in acolored de-Bruijn graph.

### 2.1

### Phylogenetic splits

In the following, we briefly recapitulate some notions and statements from the split decomposition theory introduced by Bandelt and Dress [2], and put them into context.

Definition 1 (Unordered split) Given a set O, if for two subsets A, B ⊆ O, both

A∩B =∅andA∪B=O, then the unordered pair{A, B}is a bipartitionor (unordered)

split ofO. If either Aor B is empty, a split is called trivial.

We extend the above commonly used terminology of (unordered) splits to ordered

splits—a central concept in our approach.

Definition 2 (Ordered split) If{A, B} is an unordered split ofO, the ordered pairs

(A, B)and (B, A)are ordered splits. (B, A) is called the inverse (split) of (A, B)and

vice versa.

Note that one unordered split {A, B} = {B, A} corresponds to two ordered splits

(A, B)6= (B, A). Our method will first infer ordered splits and their inverse, which will

then be combined to form unordered splits. If clear from the context, we may denote

an ordered split (A, B) by simplyA.

A set of splits S may be supplemented with weights w : S −→ R, e.g., in [2],

splits are weighted by a so-called isolation index. Strong relations between metrics

and sets of weighted unordered splits have been shown. In particular, one can

P

{A,B}∈Sw({A, B})δA(a, b) where δA(a, b) := 1 if either a or b in A, but not both,

and δA(a, b) := 0 otherwise, i.e., the weights of all splits which separate a from b are

accumulated. A set of splitsS is of the form S=Sd for some metricdif and only if it

isweakly compatible in the following sense.

Definition 3 (Weak compatibility [2]) A set of unordered splitsS onO is weakly

compatible if for any three splits {A1, B1},{A2, B2}, {A3, B3} ∈ S, there are no

ele-mentsa, a1, a2, a3∈O with{a, a1, a2, a3} ∩Ai={a, ai}fori= 1,2,3.

As a peculiarity of our approach is being not distance-based, we mention the above

relation of weakly compatible splits and distances only for the sake of completeness. We will make use the above property to filter a general set of splits such that it can be displayed as a—in most cases planar—network.

For a tree metric (also called additive metric) d, a set of splits S is of the form

S=Sd, if and only if it iscompatible in the following sense.

Definition 4 (Compatibility [2]) A set of unordered splits S onO is compatible if

for any two splits {A, B} and {A0, B0}, one of the four intersections A∩A0, A∩B0,

B∩A0, andB∩B0 is empty.

We will make use of the implied one to one correspondence of edges in a tree and

compatible splits: an edge of length w whose removal separates a tree into two trees

with leaf setsAandB, respectively, corresponds to a split{A, B} of weightw.

### 2.2

### Colored de-Bruijn graphs

A string sis a sequence of characters over a finite, non-empty set, calledalphabet. Its

length is denoted by |s|, the character at position i by s[i], and the substring from

positionithroughj bys[i..j]. A substring of lengthkis calledk-mer.

We consider agenome as a set of strings over the DNA-alphabet {A, C, G, T}. The

reverse complement of a string s is s := s[|s|]· · ·s[1], where A := T, C := G, G :=

C, T :=A.

An abstract data structure that is often used to efficiently store and process a

col-lection of genomes is the colored de-Bruijn graph (C-DGB) [9]. It is a node-labeled

graph (V, E, col) where each vertex v ∈ V represents a k-mer associated with a set

of colors col(v) representing the set of genomes the k-mer occurs in. A directed edge

from v to v0 exists if and only if for the correspondingk-mers xand x0, respectively,

x[2..k] =x0[1..k−1]. We call a path p=v1, . . . , vl of length |p|=l in a C-DBG

non-branching if all contained vertices have an in- and outdegree of one with the possible

exception ofv1 having an arbitrary indegree andvlhaving an arbitrary outdegree, and

it has the same set of colors assigned to all its vertices. A maximal non-branching path

is aunitig. In acompacted C-DBG, all unitigs are merged into single vertices.

In practice, since a genomic sequence can be read in both directions, and the actual direction of a given sequence is usually unknown, a string and its reverse complement

are assumed equivalent. Thus, in many C-DBG implementations, both ak-mer and its

reverse complement are represented by the same vertex. In the following, we will assume this being internally handled by the data structure.

### 3

### Method

The basic idea of our new approach is that a sequence which is contained as substring in

Algorithm 1SANS: Symmetric, Alignmet-free phylogeNomic Splits

INPUT: List of genomes G OUTPUT: Weighted splits over G

T := empty trie // initialize T[S] := (0,0) on first access by S C-DBG := colored de-Bruijn graph of G

foreach unitig U in C-DBG:

S := color list of U (sublist of G)

// add ordered split S or its inverse G\S to trie

if |S| < |G|/2 or ( |S| == |G|/2 and S[0] == G[0] ) then: increase first element of T[S] by length of U

else:

increase second element of T[G\S] by length of U foreach entry S in T with values (w,w’):

output unordered split {S,G\S} of weight sqrt(w*w’)

as a signal thatA should be separated fromG\A in the phylogeny. The more of those

sequences exist and the longer they are, the stronger is the signal for separation. To efficiently extract common sequences, we first construct a C-DBG of all given

genomes. Then, we collect all separation signals as ordered splits, where any unitigu

contributes|u|to the weight of an ordered splitcol(u). Since both an ordered split (A, B)

and its inverse (B, A) indicate thatAand B should be separated in the phylogeny, we

combine them to one unordered split{A, B}with an overall weight that is a combination

of the individual weights. The individual steps will be explained in more detail next.

C-DBG

Among several available implementations of C-DGBs (e.g. [1, 7, 9, 12]), we decided to use

Bifrost (Paul Melsted and Guillaume Holley,https://github.com/pmelsted/bifrost)

for the following reasons: it is easy to install and use; it is efficiently implemented; it can process full genome sequences, assemblies, read data or even combinations of these; for

read data as input, it offers some basic assembly-like filtering ofk-mers; and it realizes

a compacted C-DBG and provides a C++ API such that a traversal of the unitigs could be easily and efficiently implemented—only unitigs with heterogeneous color sets had to be split, because colors are not considered during compaction.

Accumulating split weights

Because many splits are overlapping, we use a trie data structure to store a split (as key) as path from the root to a terminal vertex, along with its weight (as value) assigned

to the terminal vertex. We represent the set of genomes G as a list with some fixed

order, and any subset ofGas sublist ofG, i.e., with the same relative order. For a split

(A, B) and its inverse (B, A), we take as key the shorter ofAand B, breaking ties by

selecting that split containingG[0], and as value the pair of weights (w, w0), wherewis

the accumulated weight of the key, andw0 the accumulated weight of its inverse. When

the trie is accessed for a key the first time, the value is initialized with (0,0).

The overall method SANS is shown in Algorithm 1, the very last step of which will be motivated in the following.

Combining splits and their inverses

To combine an ordered split (A, B) of weightwA and its inverse (B, A) of weightwB, a

naive argument would be: both indicate the same separation, so they should be taken

into account equivalently, and thus take the sum wA+wB or arithmetic mean (wA+

wB)/2. However, in our evaluation, this weighting scheme often assigned higher weight

to wrong splits than to correct splits (compared to reliable reference trees; exemplified

in Section 4.1). Instead, we revert the above argument: consider a mutation on a

(true) phylogenetic branch separating the set of genomes into subgroupsAandB. The

corresponding two variants of the affected segment will induce two unitigs with color

setsAandB, respectively. Under the infinite sites assumption, these unitigs would not

be affected by other events. So, each mutation on a branch in the phylogeny contributes

to both splits (A, B) and (B, A). We hence take the geometric mean √wA·wB such

that in case of asymmetric splits, the lower weight diminishes the total weight, and only symmetric splits receive a high overall weight.

Considering different scenarios that would affect the observation of common sub-strings in the C-DBG, some of which are illustrated in Figure 1, we observe beneficial

behavior of the weighting scheme in almost all cases: A single nucleotide variation

would cause a bubble in the C-DBG composed of two unitigs of similar lengthkeach—a

symmetric scenario in accordance with the above weighting scheme. Both aninsertion

or deletionof lengthlwould cause an asymmetric bubble and thus asymmetric weights

k−1 andl+k−1. Here, the geometric mean has the positive effect to weaken the

im-pact of the length of the event on the overall split weight. For both atransposition or

inversionof arbitrary length, the color set of the segment itself remains the same, and

only those k-mers spanning the breakpoint regions would be affected, inducing

sym-metric bubbles in accordance with the weighting scheme. Lateral gene transfer is

challenging phylogenetic reconstruction, because a subsequence of length l that is

con-tained in both the groupAcontaining the donor genome as well as the target genomeb

from the other genomesB:=G\Acan easily be misinterpreted as a signal to separate

A∪ {b}from the remainderB\{b} instead of separatingAfrom B, where the strength

of this erroneous signal grows withl. Our approach will be affected only little: On the

one hand, the unitig corresponding to the copied subsequence has color setA∪ {b}and

thus contributes to an ordered split (A∪ {b}, B\{b}) of weight l−k+ 1. On the other

hand, because the transfer does not remove any subsequence in the donor sequence,

only those k k-mers spanning the breakpoint region will be affected, inducing a unitig

with color set B\{b} whose length is independent ofl. Missing or additional data

may arise from genomic segments that are difficult to sequence or assemble and might thus be missing in some assemblies, due to the usage of different sequencing protocols, assembly tools, or filter criteria, or simply because some input files contain plasmid or mitochondrial sequences and others do not. This does not affect our approach, because additional sequence induces unitigs and thus an ordered split, but the absence of se-quence does not induce any split, not even due to breakpoint regions, because in such cases usually whole reads, contigs or chromosomes are involved. Thus, the weight of the additional ordered split would be multiplied by zero for the absent split, resulting

in a total weight of zero. Copy number changescan only be detected if the change

is from one to two orvice versa, adding or removingk-mers spanning the juncture of

the two copies. Beyond that, because thek-mer counts are not captured, our approach

is not sensible for copy number changes.

In practice, the structure of a C-DBG is much more complex than the simplified picture we draw above. Nevertheless, using the geometric yields high accuracy of the approach compared to other methods.

AAC GTT {a, b, c, d} % . & -ACG CGT {a, b} ↔ CGC GCG {a, b} ↔ GCA TGC {a, b} ACT AGT {c, d} ↔ CTC GAG {c, d} ↔ TCA TGA {c, d} & -% . CAA TTG {a, b, c, d}

(a) Single nucleotide variation in genomesa=b= AACGCAA andc=d= AACTCAA. The induced ordered split{a, b}and its inverse{c, d}of weightk = 3 each yield a corresponding unordered split{{a, b},{c, d}}of weight√k k=k= 3.

AAC GTT {a, b, c, d} % . & -ACG CGT {a, b} ↔ CGG CCG {a, b} ↔ GG· ·CC {a, b} ↔ · · · ↔ ·CA TG· {a, b} ↔ CAC GTG {a, b} ↔ ACA TGT {a, b} ACC GGT {c, d} ←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ CCA TGG {c, d} & -% . CAA TTG {a, b, c, d}

(b) Insertion/deletion of length l = 4 (or longer, indicated by dots) in genomes a = b = AACGG· · ·CACAA and c = d = AACCAA. The induced ordered split {a, b} of weight l+k−1 = l+ 2 and its inverse {c, d}of constant weight k−1 = 2 yield a corresponding unordered split{{a, b},{c, d}}of weightp(l+k−1) (k−1) =p2(l+ 2).

AAC GTT {a, b, c, d} % . & -ACG CGT {a, b} ↔ CGG CCG {a, b} ACT AGT {c, d} ↔ CTG CAG {c, d} ←−−−−−−−−−−−−−−−−−−− −−→ & - GG· ·CC {a, b, c, d} ↔ · · · ↔ ·CA TG· {a, b, c, d} % . ←−−−−−−−−−−−−−−−−−−−−−→ CAC GTG {a, b} ↔ ACA TGT {a, b} CCC GGG {c, d} ↔ CCA TGG {c, d} & -% . CAA TTG {a, b, c, d}

(c) Inversion of length l = 4 (or longer, indicated by dots) between genomes a = b = AACGG· · ·CACAA and c = d = AACTG· · ·CCCAA. The induced ordered split {a, b}

and its inverse{c, d}of constant weight 2(k−1) = 4 each yield a corresponding unordered split

{{a, b},{c, d}}of constant weightp 2(k−1) 2(k−1) = 2(k−1) = 4. AGG CCT {a} ↔ GG· ·CC {a, b} ↔ · · · ↔ ·CA TG· {a, b} ↔ CAG CTG {a} % . & -ACG CGT {b} ↔ CGG CCG {b} CAC GTG {b} ↔ ACA TGT {b} AAC GTT {b, c, d} ←−−−−−−−−−−−−−−−−−−→ % . ACC GGT {c, d} ←→ CCA TGG {c, d} ←−−−−−−−−−−−−−−−−−−→ & - CAA TTG {b, c, d}

(d) Lateral gene transfer of length l = 4 (or longer, indicated by dots) from genome a = AGG· · ·CAG to b = AACGG· · ·CACAA but not to c = d = AACCAA. Apart from mutation-independent splits for the boundaries, and the trivial split{b}(without its in-verse), the split {a, b} of weight l−k+ 1 = l−2 and its inverse {c, d} of constant length k−1 = 2 are induced, yielding a corresponding unordered split {{a, b},{c, d}} of weight

p

(l−k+ 1) (k−1) =p

2(l−2).

Figure 1: Toy examples for different mutations within four genomes a, b, c and d to

illustrate their effect on a C-DBG withk = 3. Each vertex of the C-DBG is labelled

with both itsk-mer and the reverse complement (in arbitrary order), as well as its color

set. Due to the small value of k, the C-DBG contains edges corresponding to pairs

of overlapping k-mers that are not contained in the given strings. For the purpose of

Postprocessing

Even though the geometric mean filters out many asymmetric splits, the total number

of positively weighted splits can be many-fold higher than 2n−3, the number of edges

in a fully resolved tree for n genomes. Unfortunately, the observed distribution of

split weights did not indicate any obvious threshold to separate high-weighted splits from low-weighted noise. Nevertheless, a rough cutoff can safely be applied by keeping

only the t highest weighting splits, e.g., in our evaluation t = 10n has been used for

all datasets. Additionally, we evaluated two filtering approaches: greedy weakly, i.e.,

greedily approximating a maximum weight subset that is weakly compatible and can

thus be displayed as a network, andgreedy tree, i.e., greedily approximating a maximum

weight subset that is compatible and thus corresponds to a tree. To this end, we used the corresponding options of the software tool SplitsTree [8, 10]. As we will demonstrate in the Results section, in particular the tree filter proved to be very effective in practice.

Run time complexity

Considerngenomes of lengthO(m) each. In Bifrost, the compacted C-DBG is built by

indexing ak-mer by itsminimizer, i.e., a substring with the smallest hash value among

all substrings of lengthg in a k-mer. According to the developers of Bifrost (personal

communication), inserting ak-mer and its color takesO(4(k−g)log(n)) time in the worst

case. In practice, however, each of theO(m n)k-mers can be inserted inO(log(n)) time,

and hence, building the complete C-DGB takesO(m nlog(n)) time. While iterating over

all positions in the graph, we verify whether a unitig has to be split due to a change in

the color set. Because each of thengenomes addsO(m) color assignments to the graph,

we have to do O(m n) color comparisons in total, which does not increase the overall

complexity.

Each genome contributes to at most O(m) ordered splits. So the sum of the

cardi-nality of all ordered splits, i.e., the total length of all splits in Algorithm 1, is O(m n).

Hence, the insertion and lookups of all S in trie T takes |S| time each and O(m n) in

total, and the number of terminal vertices of T, i.e., the final number of unordered

splits, is in O(m n), too. For ease of postprocessing, splits are ordered by decreasing

weight, increasing the run time for split extraction toO(m nlog(m n)), orO(m nlog(n))

to output only thet,t∈O(n), highest weighting splits, respectively.

### 4

### Results

In this section, we present several use cases in order to exemplify robustness and different other characteristics of our approach SANS. We compare to the following other whole-genome based reconstruction tools.

MultiSpaM [3] samples a constant, high number of small, gap-free alignments of four genomes. The implied quartet topologies are combined to an overall tree topology. To the best of our knowledge, all other tools are distance-based and rely on pairwise comparisons. Interestingly, although all methods are based on lengths or numbers of common subsequences or patterns, their results differ considerably from those of SANS.

Co-phylog [16] analyses each genome in terms of certain patterns (C-grams, O-grams)

and compares their characteristics (context). In andi [6], enhanced suffix arrays are

used to detect pairs of maximal unique matches that are used to anchor ungapped local

alignments, based on which pairwise distances are computed. CVTree3 [20] correctsk-,

k−1, andk−2-mer counts by subtracting random background of neutral mutations using

a (k−2)-th Markov assumption. In FSWM [11], matches of patterns including match

0 10 20 30 40 50 1e+04 1e+05 1e+06 1e+07 1e+08 rank split w

eight (log scale)

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arithmetic mean, correct arithmetic mean, false geometric mean, correct geometric mean, false

(a) Comparison of accuracy for using arith-metic or geometric mean for combining weights of splits and their inverse each. Splits have been sorted by the combined weight and the 50 highest weighting splits are shown. Color indicates whether a split agrees with the reference [15]. ana mel sec sim ere yak gri vir moj wil per pse

(b) Visualization of greedily extracted weakly compatible subset of splits using SplitsTree [8, 10]. As by default, geometric mean has been used for combining weights of splits and their inverse each.

Figure 2: Reconstructed phylogenetic splits on the Drosophila dataset [15].

Unless stated otherwise, a k-mer length of 31 has been used for constructing the

C-DBG (Bifrost default) for SANS. Accuracy has been measured in terms of topological Robinson-Foulds distance, i.e., a predicted edge or split is correct if and only if the reference tree contains an edge that separates the same two sets of leaves. All tools have been run on a single 2 GHz processor and times are given in CPU hours (user time).

### 4.1

### Drosophila

This dataset comprises assemblies from 12 species of the genusdrosophilaobtained from

the database FlyBase (flybase.org, latest release before Feb. 2019 ofall-chromosome

-files each) [15].

Although being “simple” in the sense that it contains only a small number of genomes, its analysis exemplifies the following aspects: (i) The effectiveness of our method for medium sized input files: for a total of more than 2 161 Mbp (180 Mbp on average), SANS inferred the correct tree within 168 minutes and using up to 25 GB

of memory. We ran CVTree3 with various values of k. In the best cases (k= 12 and

13), 7 of 9 internal edges have been inferred correctly taking 95 and 162 minutes, and

up to 26 and 87 GB of memory, respectively. (Fork = 11, only 4 internal edges were

correct, and for k > 13, the computation ran out of memory.) Both Co-phylog and

FSWM did not finish within 48 hours, and both MultiSpaM and andi could not pro-cess this dataset sucpro-cessfully. (ii) As can be seen in Figure 2a, the tendency of correct splits having a high weight is stronger when combining splits and their inverse using the geometric mean than using the arithmetic mean. (iii) Even though the reconstruction shown in Figure 2b contains 45 splits—in comparison to 21 edges in a binary tree—, the visualization is close to a tree structure.

0 50 100 150 200 0 2 4 6 8 10 #assemblies time (hours) ● ● ● ● ● FSWM Co−phylog andi SANS

(a) Running time for computing phylogenies on random subsamples. Times for SANS in-clude DBG construction with k = 31, split extraction and agglomeration.

0.5 0.6 0.7 0.8 0.9 1.0 0.5 0.6 0.7 0.8 0.9 1.0 recall precision ●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ● ● ● ● k=21 k=31 k=45 k=63 greedy tree greedy weakly Co−phylog FSWM andi

(b) For different values ofk, weakly compat-ible subsets (triangles) and a trees (bullets) have been greedily extracted. For varying val-ues of i, only the i highest weighting splits have been considered as “positives” to deter-mine precision and recall each (lines).

Figure 3: Comparison of running time and accuracy of different methods on the ParaC

dataset [18] comprising assemblies ofn= 220 genomes.

### 4.2

### Salmonella enterica

### Para C

This dataset is of special interest as the contained assemblies from 220 genomes of

different serovars within the Salmonella enterica Para C lineage include that of an

ancient Paratyphi C genome obtained from 800 year old DNA [18], the placement of which is especially difficult due to missing data. As reference, we consider a maximum-likelihood based tree on nonrecombinant SNP data [18, Figure 5a].

We studied the running time behaviour of the different methods for random subsam-ples of increasing size. As shown in Figure 3a, for this high number of closely related genomes, we observed a super-linear running time of up to 41 minutes for andi, about 5 hours for Co-phylog, and up to 43 hours for FSWM, whereas the reconstruction of SANS shows a linear increase (Pearson correlation coefficient 0.9994) to about 10 minutes. The memory requirement of both SANS and Co-phylog remained below 0.5 GB, whereas andi required about 1 GB, and FSWM required up to about 17 GB. We ran CVTree3 with

ten values of kbetween 5 and 27, but none of the resulting trees contained more than

5 correct internal edges. For MultiSpaM, we increased the number of sampled quartets

from the default of 106to up to 108, which increased the running time from about one

hour to about 66 hours. Both recall and precision improved but were still below 0.2 for internal edges.

The accuracy of the reconstructions with respect to the reference is visualized in Figure 3b. In particular, we observe: (i) the split reconstruction by SANS and the tree inferred by Co-phylog are comparably accurate and both are more accurate than the FSWM and andi tree, (ii) greedily extracting high weighting splits to filter for a tree selects correct splits while discarding false splits with very high precision, (iii) greedily extracting high weighting splits to filter for a weakly compatible subset also selects correct splits, but, as expected, has a lower precision as the tree filter, because more splits are kept than there are edges in a tree, and (iv) the results of SANS are robust

0 500 1000 1500 0 50 100 150 200 250 300 #assemblies time (min) ● ● ● ● ● ● ● ● ● ● 0 20 40 60 80 peak memor y (GB) ● ● time memory time C−DBG memory C−DBG

(a) Running time and peak memory usage of SANS. Values including C-DBG construction, split extraction and agglomeration, as well as C-DBG construction only are given.

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 recall precision ● ● ● ● ● ● / / / / / / /

greedy tree, 250 assemblies greedy tree, 500 assemblies greedy tree, 1000 assemblies greedy tree, 1500 assemblies Co−phylog, 250 assemblies FSWM, 250 assemblies andi, 250 assemblies (all edges / internal only)

(b) Accuracy with respect to the reference phylogeny [19, Fig. 2A].

Figure 4: Efficiency and accuracy on the Salmonella enterica dataset [19]. Values have been averaged over processing two random subsamples each.

### 4.3

### Salmonella enterica

### subspecies

### enterica

In comparison to the ParaC dataset, the 2 964 genomes studied by Zhou et al. [19]

are not only a larger but also a more diverse selection of Salmonella enterica strains.

As reference, we consider a maximum-likelihood based tree on 3 002 concatenated core genes [19, Figure 2A, supertree 3].

The probability to observe long k-mers that are conserved in such a high number

of more diverse genomes is lower than for the previous datasets. Hence, we selected a

smaller k-mer length of k = 21. To assess the efficiency and accuracy for increasing

number of genomes, we sampled subsets of up to 1 500 assemblies. To process the

smallest considered subsample of size 250, andi took about 110 minutes, whereas Co-phylog and FSWM took already more than 9 and 50 hours, respectively, and MultiSpam

was not able to process this dataset at all. We ran CVTree3 with all values ofkbetween

6 and 14, but in the best case (k = 8), the resulting tree contained only 33 (of 247)

correct internal edges such that we did not further consider CVTree3 in our evaluation. The memory usage for split extraction and agglomeration clearly dominates those of the C-DBG construction by Bifrost such that processing the complete dataset was not possible with our current implementation of SANS. Figure 4a shows a slightly super-linear runtime and memory consumption of up to about 300 minutes and 80 GB for processing 1 500 assemblies. As can be seen in Figure 4b, both precision and recall vary only slightly for this wide range of input size. Keeping in mind that a final split of high

weight strictly requires the observation ofboth unordered pairs, this is a quite promising

result for this first investigation of the methodology. In particular, whereas for distance-based methods, all leaf-edges are inferred by construction and can never be false, a trivial split separating a leaf from the remaining tree, requires not only some sequence unique

to the leaf but also sequence that is contained in all othern−1 genomes. Also note that

measuring accuracy by counting correct and false splits corresponding to the topological Robinson-Foulds distance has to be interpreted with care. A single misplaced leaf breaks all splits between its correct and actual location. However, this is a desired behaviour in this context, because, in a phylogeny of several hundred genomes, each genome should at least be located in the correct area, whereas the complete misplacement even of a single genome can easily lead to wrong biological conclusions.

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 recall precision ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ● ● ● ● / / / / / / SANS

SANS, greedy tree CVTree3 (k=8) co−phylog FSWM MultiSpaM (w.r.t. ref. 1 / ref. 2)

(a) Accuracy of different tools w.r.t. two reference trees [5, Figures 3 and 4] shown in blue and red, respectively. For SANS, for varying values of i, only the i high-est weighting splits have been considered as “positives” to determine precision and recall each. AR158.fa MpV-12T.fa OlV1.fa OtV2.fa OtV1.fa OtV5.fa OtV6.fa MpV1.fa MpV-PL1.fa MpV-SP1.fa BpV2.fa BpV1.fa PBCV1.fa

(b) Visualization of greedily extracted weakly compatible subset of splits using SplitsTree [8, 10].

Figure 5: Reconstruction results on the prasinovirus dataset [5].

### 4.4

### Prasinoviruses

Viral genomes are short and highly diverse—posing the limits of phylogenetic recon-struction based on sequence conservation. Here we consider complete genomes of 13 prasinoviruses, which are relatively large (213 Kbp on average) [5]. As references, we consider two trees reported in the original study, one of which is based on the pres-ence and abspres-ence of shared putative genes [5, Figure 3], and the other is a maximum likelihood estimation based on a marker gene (DNA polymerase B) [5, Figure 4].

Due to the small size of the input, it could be processed by all tools, where time and memory consumption were negligible. Only andi could not process this dataset successfully (“very little homology was found”). Results are shown in Figure 5a. The visualization of the predicted splits in Figure 5b exemplifies the explanatory power of the split framework. While main separations supported by both reference trees are recognizable as strong splits in the net, separations in which the two reference trees disagree are also shown as weakly compatible splits.

### 4.5

### Vibrio cholerae

The dataset comprises 22 genomes from the species Vibrio cholerae, 7 of which have

been sequenced from clinical samples and are labelled “pandemic genome” (PG), and the remaining 15 have been sequenced from non-clinical samples and are labelled “en-vironmental genome” (EG) [14, primary dataset]. As already observed in the original study, for these genomes, it is difficult to reconstruct a reliable, fully resolved tree. Nev-ertheless, representing the phylogeny in form of splits shows a strong separation of the pandemic from the environmental group. The phylogeny presented by the authors of the original study [14, Supplementary Figure 1a] is based on 126 099 sites extracted from alignment blocks.

Comparing our reconstruction results to the reference, both shown in Figure 6, we make two observations. (i) Our reconstruction also separates the pandemic from the

N16961
O395
GBE0428
GBE1173
GBE1114
TMA21
62339
HE48
CT536993
HE09
VL426
RC385
GBE1068
GBE0658
12129
LMA38944
Bgd8
MQ1795
MJ1236
Bgd1
Bgd5
**PG**
**EG**

(a) Visualization of greedily extracted weakly compat-ible subset of splits. For taxa highlighted in bold, only read data was available on NCBI (input option -s of Bifrost has been used); for Taxon TM1107980, no data was available on NCBI (February 2019).

PG EG 12129 LMA38944 CT536993 GBE0658 GBE1068 RC385 albensisVL426 HE09 TM1107980 HE48 GBE1173 GBE1114 TMA21 62339 GBE0428 Bgd1 MQ1795 MJ1236 O395 N16961Bgd5 Bgd8 MQ1795 N16961 MJ1236 O395 Bgd1 Bgd5 Bgd8 12129 LMA38944 CT536993 GBE0658 GBE1068 RC385 albensisVL426 HE09 TM1107980 HE48 GBE1114 GBE1173 TMA21 62339 GBE0428

(b) Reference phylogeny. Figure reprinted from Shapiro et al. [14, Supplementary Figure 1a].

Figure 6: Splits reconstructed for theV. cholerae dataset [14] by SANS (left) and by

Shapiroet al.[14] (right) visualized with SplitsTree [8, 10].

collecting the sequence data, for some of the genomes, we found assemblies, whereas for others, only read data was available. Because the used C-DBG implementation Bifrost supports a combination of both types as input, we were able to reconstruct a joint phylogeny without extra effort or obvious bias in the result.

### 5

### Discussion and Outlook

We proposed a new k-mer based method for phylogenetic inference that neither relies

on alignments to a reference sequence nor on pairwise or multiple alignments to infer markers. Prevailing whole-genome approaches perform pairwise comparisons to deter-mine a quadratic number of distances to finally infer a linear number of tree edges. In contrast, in our approach, the length of conserved sequences is extracted from a colored de-Bruijn graph to first infer signals for phylogenetic sub-groups. These signals are then combined with a symmetry assumption to weighted phylogenetic splits. Evaluations on several real datasets have proven comparable or better efficiency and accuracy compared

to other whole-genome approaches. Our results indicate robustness in terms of k-mer

length, as well as the taxonomic order, size and number of the genomes. The analysis of a dataset composed of both assembly and read data indicated also robustness in this regard—an important feature, which we want to investigate further.

A distinctive feature of the proposed methodology is the direct association of a phylogenetic split to the conserved subsequences it has been derived from, which is not possible for distance-based methods. We plan to enrich our implementation with this valuable possibility to allow the analysis of characteristic subsequences of identified subgroups, or subsequences inducing phylogenetic splits off the main tree, e.g. horizontal gene transfer. Here, the applied generalization of trees plays an important role, e.g., circular split systems are more strict than weakly compatible sets and might thus be a promising alternative to be studied further.

At its current state, apart from iterating a colored de-Bruijn graph and agglomerating unitig lengths, the only elaborate ingredient so far is the symmetry assumption realized by applying the geometric mean. We believe that the general approach still harbors much potential to be further refined by, e.g., statistical models, advanced data structures, pre-or postprocessing, to further increase its accuracy and efficiency.

### Acknowledgements

I thank Guillaume Holley for support on Bifrost, Nina Luhmann for pointers to data sets, and Andreas Rempel for programming assistance.

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